Frequency combs are having a broad impact on science and technology because they provide a way to coherently link radio/microwave-rate electrical signals with optical-rate signals derived from lasers and atomic transitions. A new, miniature realization, the microcomb, that uses chip-based microresonators can potentially revolutionize instrumentation, time keeping, spectroscopy, and navigation. Microcombs were first demonstrated using a form of cascaded four-wave mixing. However, the recent discovery of dissipative soliton microcombs enables phase-locked spectra with reproducible envelopes, as required in many frequency comb applications. In addition, these solitons are confined in a high-Q microresonator, thereby creating a rich landscape for research in nonlinear optical phenomena. In this thesis, these solitons are demonstrated for the first time in a silica microcavity. Significantly, the device provides a microwave-detectable soliton repetition rate, which is essential to many comb applications. The unusual properties of the solitons are studied from a theoretical viewpoint using a Lagrangian formalism and predictions of the theory are confirmed experimentally. In the course of this work, a new optical soliton, the Stokes soliton, was also discovered. In addition to soliton mode locking, another novel and compact platform, the electro-optical modulation frequency comb, was studied. This type of frequency comb was used to demonstrate a novel electro-optic form of frequency division for stable microwave synthesis. It was also modified to perform astronomical calibration for exoplanet detection at the Keck Observatory in Hawaii.

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Advisor and committee chair names found in the thesis' metadata record in the digital repository.